Convexity via nonnegative second derivative

A twice differentiable function is convex iff f''≥0 on the interval

Convexity via nonnegative second derivative

Corollary. Let $f:I\to\mathbb{R}$ be twice differentiable on a nonempty open interval $I\subset\mathbb{R}$ . Then $f$ is convex on $I$ if and only if

f''(x)\ge 0 \quad \text{for all }x\in I.

Connection. This follows from convexity ⇔ monotone derivative , since $f''\ge 0$ is equivalent to $f'$ being nondecreasing.